Infinite dimensional Morse theory and multiple solution problems

✍ Scribed by K.C. Chang

• Year: 1992
• Tongue: English
• Leaves: 324
• Edition: 1
• Category: Library

✦ Synopsis

This advanced graduate level monograph is the first to present a wide variety of critical point theorems in a unified framework. The author not only employs Morse theory as a tool to study multiple solutions to differential equations arising in the calculus of variations, but covers a broad range of applications to semilinear elliptic PDE, to dynamical systems and symplectic geometry, and to geometry of harmonic maps and minimal surfaces. Critical groups for isolated critical points or orbits - which provide more information than the Leray-Schauder index - are introduced. Topics covered include basic Morse theory and its various extensions, minimax principles in Morse theory, and applications of semilinear boundary value problems, periodic solutions of Hamiltonian systems, and harmonic maps. In a self-contained appendix, the author presents Witten's proof of Morse inequalities. Containing several new results, this volume will be attractive and germaine to researchers and graduate students working in nonlinear analysis, nonlinear functional analysis, partial differential equations, ordinary differential equations, differential geometry, and topology. For research mathematicians, physicists and graduate students.

✦ Table of Contents

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Contents......Page 6
Preface......Page 8
Introduction......Page 10
1. A Review of Algebraic Topology......Page 12
2. A Review of the Banach-Finsler Manifold......Page 25
3. Pseudo Gradient Vector Field and the Deformation Theorems......Page 30
4. Critical Groups and Morse Type Numbers......Page 43
5. Gromoll-Meyer Theory......Page 54
6. Extensions of Morse Theory......Page 65
6.1. Morse Theory Under General Boundary Conditions......Page 66
6.2. Morse Theory on a Locally Convex Closed Set......Page 71
7. Equivariant Morse Theory......Page 76
7.1. Preliminaries......Page 77
7.2. Equivariant Deformation......Page 78
7.3. The Splitting Theorem and the Handle Body Theorem for Critical Manifolds......Page 80
7.4. G-Cohomology and G-Critical Groups......Page 85
1. Topological Link......Page 94
2.1. Link......Page 103
2.2. Genus and Cogenus......Page 107
3.1. Degree theory......Page 110
3.2. Ljusternik-Schnirelman Theory......Page 116
3.3. Relative Category......Page 120
4. Invariant Functionals......Page 122
5. Some Abstract Critical Point Theorems......Page 132
6.1. Perturbation on Critical Manifolds......Page 142
6.2. Uhlenbeck's Perturbation Method......Page 147
1. Preliminaries......Page 151
2. Superlinear Problems......Page 155
3.1. Nonresonance and Resonance with the Landesman-Lazer Condition......Page 164
3.2. Strong Resonance......Page 167
3.3. A Bifurcation Problem......Page 172
3.4. Jumping Nonlinearities......Page 175
3.5. Other Examples......Page 180
4.1. Functionals Bounded From Below......Page 183
4.2. Oscillating Nonlinearity......Page 184
4.3. Even Functionals......Page 187
4.4. Variational Inequalities......Page 188
1. Asymptotically Linear Systems......Page 193
2.1. Saddle Point Reduction......Page 199
2.2. A Multiple Solution Theorem......Page 206
2.3. Periodic Nonlinearity......Page 209
3. Singular Potentials......Page 214
4. The Multiple Pendulum Equation......Page 220
5.1. Conjectures......Page 226
5.2. The Fixed Point Conjecture on (T2s,wo)......Page 229
5.3. Lagrange Intersections for (CP", RP")......Page 231
1. Harmonic Maps and the Heat Flow......Page 240
2. The Morse Inequalities......Page 257
3. Morse Decomposition......Page 261
4. The Existence and Multiplicity for Harmonic Maps......Page 268
5. The Plateau Problem for Minimal Surfaces......Page 271
Appendix: Witten's Proof of the Morse Inequalities......Page 0
1. A Review of Hodge Theory......Page 285
2. The Witten Complex......Page 293
3. Weak Morse Inequalities......Page 298
4. Morse Inequalities......Page 306
References......Page 309
Index of Notation......Page 321
Index......Page 322
Back Cover......Page 324